So,we have to find a recurrence relation for the number of sequences of yellow and purple stones of length \(\displaystyle{N}\), where, each sequence has the property that no two adjacent stones are purple.

Let \(\displaystyle{a}_{{N}}\) be the number of such sequences of length \(\displaystyle{N}\).

There are two cases:

1) The last stone is yellow.

In this case, the first \(\displaystyle{n}-{1}\) stones must not have 2 adjacent purples.

Thus, there are \(\displaystyle{a}_{{{N}−{1}}}\) such sequences.

2)The last stone is purple.

In this case, the last 2 stones must be yellow and purple. So, the first \(\displaystyle{N}-{2}\) stones must not have 2 adjacent purples.

Thus, there are \(\displaystyle{a}_{{{N}−{2}}}\) such sequences.

In total, there are \(\displaystyle{a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\) sequences.

Therefore, \(\displaystyle{a}_{{N}}={a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\), here, \(\displaystyle{a}_{{0}}={1}\), \(\displaystyle{a}_{{1}}={2}\). Hence, the required recurrence relation is \(\displaystyle{a}_{{N}}={a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\)

Let \(\displaystyle{a}_{{N}}\) be the number of such sequences of length \(\displaystyle{N}\).

There are two cases:

1) The last stone is yellow.

In this case, the first \(\displaystyle{n}-{1}\) stones must not have 2 adjacent purples.

Thus, there are \(\displaystyle{a}_{{{N}−{1}}}\) such sequences.

2)The last stone is purple.

In this case, the last 2 stones must be yellow and purple. So, the first \(\displaystyle{N}-{2}\) stones must not have 2 adjacent purples.

Thus, there are \(\displaystyle{a}_{{{N}−{2}}}\) such sequences.

In total, there are \(\displaystyle{a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\) sequences.

Therefore, \(\displaystyle{a}_{{N}}={a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\), here, \(\displaystyle{a}_{{0}}={1}\), \(\displaystyle{a}_{{1}}={2}\). Hence, the required recurrence relation is \(\displaystyle{a}_{{N}}={a}_{{{N}−{1}}}+{a}_{{{N}−{2}}}\)